String topology

String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by Moira Chas and Dennis Sullivan (1999).

Motivation

While the singular cohomology of a space has always a product structure, this is not true for the singular homology of a space. Nevertheless, it is possible to construct such a structure for an oriented manifold ${\displaystyle M}$ of dimension ${\displaystyle d}$. This is the so-called intersection product. Intuitively, one can describe it as follows: given classes ${\displaystyle x\in H_{p}(M)}$ and ${\displaystyle y\in H_{q}(M)}$, take their product ${\displaystyle x\times y\in H_{p+q}(M\times M)}$ and make it transversal to the diagonal ${\displaystyle M\hookrightarrow M\times M}$. The intersection is then a class in ${\displaystyle H_{p+q-d}(M)}$, the intersection product of ${\displaystyle x}$ and ${\displaystyle y}$. One way to make this construction rigorous is to use stratifolds.

Another case, where the homology of a space has a product, is the (based) loop space ${\displaystyle \Omega X}$ of a space ${\displaystyle X}$. Here the space itself has a product

${\displaystyle m\colon \Omega X\times \Omega X\to \Omega X}$

by going first the first loop and then the second. There is no analogous product structure for the free loop space ${\displaystyle LX}$ of all maps from ${\displaystyle S^{1}}$ to ${\displaystyle X}$ since the two loops need not have a common point. A substitute for the map ${\displaystyle m}$ is the map

${\displaystyle \gamma \colon {\rm {Map}}(S^{1}\lor S^{1},M)\to LM}$

where ${\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)}$ is the subspace of ${\displaystyle LM\times LM}$, where the value of the two loops coincides at 0 and ${\displaystyle \gamma }$ is defined again by composing the loops.

The Chas–Sullivan product

The idea of the Chas–Sullivan product is to now combine the product structures above. Consider two classes ${\displaystyle x\in H_{p}(LM)}$ and ${\displaystyle y\in H_{q}(LM)}$. Their product ${\displaystyle x\times y}$ lies in ${\displaystyle H_{p+q}(LM\times LM)}$. We need a map

${\displaystyle i^{!}\colon H_{p+q}(LM\times LM)\to H_{p+q-d}({\rm {Map}}(S^{1}\lor S^{1},M)).}$

One way to construct this is to use stratifolds (or another geometric definition of homology) to do transversal intersection (after interpreting ${\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)\subset LM\times LM}$ as an inclusion of Hilbert manifolds). Another approach starts with the collapse map from ${\displaystyle LM\times LM}$ to the Thom space of the normal bundle of ${\displaystyle {\rm {Map}}(S^{1}\lor S^{1},M)}$. Composing the induced map in homology with the Thom isomorphism, we get the map we want.

Now we can compose ${\displaystyle i^{!}}$ with the induced map of ${\displaystyle \gamma }$ to get a class in ${\displaystyle H_{p+q-d}(LM)}$, the Chas–Sullivan product of ${\displaystyle x}$ and ${\displaystyle y}$ (see e.g. Cohen & Jones (2002)).

Remarks

• As in the case of the intersection product, there are different sign conventions concerning the Chas–Sullivan product. In some convention, it is graded commutative, in some it is not.
• The same construction works if we replace ${\displaystyle H}$ by another multiplicative homology theory ${\displaystyle h}$ if ${\displaystyle M}$ is oriented with respect to ${\displaystyle h}$.
• Furthermore, we can replace ${\displaystyle LM}$ by ${\displaystyle L^{n}M={\rm {Map}}(S^{n},M)}$. By an easy variation of the above construction, we get that ${\displaystyle {\mathcal {}}h_{*}({\rm {Map}}(N,M))}$ is a module over ${\displaystyle {\mathcal {}}h_{*}L^{n}M}$ if ${\displaystyle N}$ is a manifold of dimensions ${\displaystyle n}$.
• The Serre spectral sequence is compatible with the above algebraic structures for both the fiber bundle ${\displaystyle {\rm {ev}}\colon LM\to M}$ with fiber ${\displaystyle \Omega M}$ and the fiber bundle ${\displaystyle LE\to LB}$ for a fiber bundle ${\displaystyle E\to B}$, which is important for computations (see Cohen, Jones & Yan (2004) harvtxt error: no target: CITEREFCohenJonesYan2004 (help) and Meier (2010) harvtxt error: no target: CITEREFMeier2010 (help)).

The Batalin–Vilkovisky structure

There is an action ${\displaystyle S^{1}\times LM\to LM}$ by rotation, which induces a map

${\displaystyle H_{*}(S^{1})\otimes H_{*}(LM)\to H_{*}(LM)}$.

Plugging in the fundamental class ${\displaystyle [S^{1}]\in H_{1}(S^{1})}$, gives an operator

${\displaystyle \Delta \colon H_{*}(LM)\to H_{*+1}(LM)}$

of degree 1. One can show that this operator interacts nicely with the Chas–Sullivan product in the sense that they form together the structure of a Batalin–Vilkovisky algebra on ${\displaystyle {\mathcal {}}H_{*}(LM)}$. This operator tends to be difficult to compute in general. The defining identities of a Batalin-Vilkovisky algebra were checked in the original paper "by pictures." A less direct, but arguably more conceptual way to do that could be by using an action of a cactus operad on the free loop space ${\displaystyle LM}$.[1] The cactus operad is weakly equivalent to the framed little disks operad[2] and its action on a topological space implies a Batalin-Vilkovisky structure on homology.[3]

Field theories

The pair of pants

There are several attempts to construct (topological) field theories via string topology. The basic idea is to fix an oriented manifold ${\displaystyle M}$ and associate to every surface with ${\displaystyle p}$ incoming and ${\displaystyle q}$ outgoing boundary components (with ${\displaystyle n\geq 1}$) an operation

${\displaystyle H_{*}(LM)^{\otimes p}\to H_{*}(LM)^{\otimes q}}$

which fulfills the usual axioms for a topological field theory. The Chas–Sullivan product is associated to the pair of pants. It can be shown that these operations are 0 if the genus of the surface is greater than 0 (see Tamanoi (2010))

A more structured approach (exhibited in Godin (2008)) gives ${\displaystyle {\mathcal {}}(H_{*}(LM),H_{*}(LM))}$ the structure of a degree ${\displaystyle d}$ open-closed homological conformal field theory (HCFT) with positive boundary. Ignoring the open-closed part, this amounts to the following structure: let ${\displaystyle S}$ be a surface with boundary, where the boundary circles are labeled as incoming or outcoming. If there are ${\displaystyle p}$ incoming and ${\displaystyle q}$ outgoing and ${\displaystyle n\geq 1}$, we get operations

${\displaystyle H_{*}(LM)^{\otimes p}\to H_{*}(LM)^{\otimes q}}$

parametrized by a certain twisted homology of the mapping class group of ${\displaystyle S}$.

References

1. Voronov, Alexander (2005). "Notes on universal algebra". Graphs and Patterns in Mathematics and Theoretical Physics (M. Lyubich and L. Takhtajan, eds.). Providence, RI: Amer. Math. Soc. pp. 81–103.
2. Cohen, Ralph L.; Hess, Kathryn; Voronov, Alexander A. (2006). "The cacti operad". String topology and cyclic homology. Basel: Birkhäuser. ISBN 978-3-7643-7388-7.
3. Getzler, Ezra (1994). "Batalin-Vilkovisky algebras and two-dimensional topological field theories". Comm. Math. Phys. 159 (2): 265–285. arXiv:hep-th/9212043.

• Chas, Moira; Sullivan, Dennis (1999). "String Topology". arXiv:math/9911159v1.
• Cohen, Ralph L.; Jones, John D. S. (2002). "A homotopy theoretic realization of string topology". Mathematische Annalen. 324: 773–798. arXiv:math/0107187. doi:10.1007/s00208-002-0362-0. MR 1942249.
• Ralph Louis Cohen, John D. S. Jones, and Jun Yan, The loop homology algebra of spheres and projective spaces in Categorical decomposition techniques in algebraic topology: International Conference in Algebraic Topology, Isle of Skye, Scotland, June 2001, Birkhäuser, p. 77–92 (2004).
• Meier, Lennart (2011). "Spectral Sequences in String Topology". Algebraic & Geometric Topology. 11 (5): 2829–2860. arXiv:1001.4906. doi:10.2140/agt.2011.11.2829. MR 2846913.
• Godin, Véronique (2008). "Higher string topology operations". arXiv:0711.4859v2.
• Tamanoi, Hirotaka (2010). "Loop coproducts in string topology and triviality of higher genus TQFT operations". Journal of Pure and Applied Algebra. 214 (5): 605–615. arXiv:0706.1276. doi:10.1016/j.jpaa.2009.07.011. MR 2577666.